| L(s) = 1 | + (0.190 − 0.211i)2-s + (0.988 − 1.42i)3-s + (0.200 + 1.90i)4-s + (1.81 + 2.01i)5-s + (−0.112 − 0.480i)6-s + (−3.72 − 1.65i)7-s + (0.904 + 0.656i)8-s + (−1.04 − 2.81i)9-s + 0.772·10-s + (1.48 − 2.96i)11-s + (2.91 + 1.60i)12-s + (−3.76 − 0.800i)13-s + (−1.06 + 0.472i)14-s + (4.65 − 0.588i)15-s + (−3.44 + 0.731i)16-s + (−1.33 + 4.10i)17-s + ⋯ |
| L(s) = 1 | + (0.134 − 0.149i)2-s + (0.570 − 0.821i)3-s + (0.100 + 0.954i)4-s + (0.810 + 0.900i)5-s + (−0.0460 − 0.196i)6-s + (−1.40 − 0.626i)7-s + (0.319 + 0.232i)8-s + (−0.348 − 0.937i)9-s + 0.244·10-s + (0.446 − 0.894i)11-s + (0.840 + 0.462i)12-s + (−1.04 − 0.222i)13-s + (−0.283 + 0.126i)14-s + (1.20 − 0.151i)15-s + (−0.860 + 0.182i)16-s + (−0.323 + 0.996i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 99 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.982 + 0.186i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 99 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.982 + 0.186i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.24185 - 0.116757i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.24185 - 0.116757i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (-0.988 + 1.42i)T \) |
| 11 | \( 1 + (-1.48 + 2.96i)T \) |
| good | 2 | \( 1 + (-0.190 + 0.211i)T + (-0.209 - 1.98i)T^{2} \) |
| 5 | \( 1 + (-1.81 - 2.01i)T + (-0.522 + 4.97i)T^{2} \) |
| 7 | \( 1 + (3.72 + 1.65i)T + (4.68 + 5.20i)T^{2} \) |
| 13 | \( 1 + (3.76 + 0.800i)T + (11.8 + 5.28i)T^{2} \) |
| 17 | \( 1 + (1.33 - 4.10i)T + (-13.7 - 9.99i)T^{2} \) |
| 19 | \( 1 + (-1.31 - 0.953i)T + (5.87 + 18.0i)T^{2} \) |
| 23 | \( 1 + (-0.932 + 1.61i)T + (-11.5 - 19.9i)T^{2} \) |
| 29 | \( 1 + (-3.24 - 1.44i)T + (19.4 + 21.5i)T^{2} \) |
| 31 | \( 1 + (-4.75 - 1.01i)T + (28.3 + 12.6i)T^{2} \) |
| 37 | \( 1 + (6.26 - 4.55i)T + (11.4 - 35.1i)T^{2} \) |
| 41 | \( 1 + (-6.27 + 2.79i)T + (27.4 - 30.4i)T^{2} \) |
| 43 | \( 1 + (0.492 + 0.853i)T + (-21.5 + 37.2i)T^{2} \) |
| 47 | \( 1 + (0.613 - 5.83i)T + (-45.9 - 9.77i)T^{2} \) |
| 53 | \( 1 + (0.485 + 1.49i)T + (-42.8 + 31.1i)T^{2} \) |
| 59 | \( 1 + (1.18 + 11.3i)T + (-57.7 + 12.2i)T^{2} \) |
| 61 | \( 1 + (-8.41 + 1.78i)T + (55.7 - 24.8i)T^{2} \) |
| 67 | \( 1 + (-0.870 + 1.50i)T + (-33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 + (1.77 - 5.46i)T + (-57.4 - 41.7i)T^{2} \) |
| 73 | \( 1 + (-3.27 + 2.37i)T + (22.5 - 69.4i)T^{2} \) |
| 79 | \( 1 + (2.86 - 3.18i)T + (-8.25 - 78.5i)T^{2} \) |
| 83 | \( 1 + (-6.35 + 1.35i)T + (75.8 - 33.7i)T^{2} \) |
| 89 | \( 1 - 9.26T + 89T^{2} \) |
| 97 | \( 1 + (4.96 - 5.50i)T + (-10.1 - 96.4i)T^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−13.72852534063002869980888444895, −12.94087372573663793684712750282, −12.13022335968667436094123412968, −10.64084234767695070322627178623, −9.527347043528805074341844997899, −8.224086350189085005093700284438, −6.91076595611357590763156873957, −6.37012252149006815699963778229, −3.55227907585128521630916855940, −2.65406891634331976416492688329,
2.44701334438455928447656043559, 4.63859282580886674961070861931, 5.57133866703025710162042988045, 6.95531433698150859266549719626, 9.155767151214540432738753439623, 9.481204265633909692569617467915, 10.17294638988924495433180745303, 11.94540667122744894762012454114, 13.15900615904667358527293991853, 13.98370254163893742876827368108
